The basic ring of a locally artinian category and applications
Let \({\mathcal C}\) be a locally artinian Grothendieck category. The authors associate to \({\mathcal C}^ a \)topological ring (B,\(\beta)\) with the following properties: 1) (B,\(\beta)\) is strictly linearly compact and B/J(B) is a direct product of division rings. 2) The category \({\mathcal C}\) is equivalent to the category \({\mathcal T}_{\beta}\) of all \(\beta\)-torsion left B-modules (this is a slight improvement of a result of \textit{P. N. Ánh} [J. Pure Appl. Algebra 36, 221-224 (1985; Zbl 0563.18006)]. The topological ring (B,\(\beta)\) is called the basic ring of \({\mathcal C}\). (B,\(\beta)\) is canonically associated to \({\mathcal C}\). In fact it is proved that two locally artinian Grothendieck categories are equivalent if and only if their basic rings are topologically isomorphic. Using these results it is proved that if \({\mathcal C}\) has an artinian generator then the basic ring (B,\(\beta)\) of \({\mathcal C}\) is discrete and hence \({\mathcal C}\) is equivalent to B-Mod. This improves a theorem of \textit{C. Năstăsescu} [Commun. Algebra 9, 1981-1996 (1981; Zbl 0481.16006)]. The general Hopkins-Levitzki theorem [obtained by \textit{C. Năstăsescu}, Rev. Roum. Math. Pures Appl. 24, 745-758 (1979; Zbl 0413.18009)] is obtained as a corollary of this result. Finally the class of linearly compact rings whose torsion class has an artinian generator is characterized.
scientific article; zbMATH DE number 3993824
scientific article